Schwinger's picture of quantum mechanics: 2-groupoids and symmetries

• URL: http://arxiv.org/abs/2104.13880v1
• Date: Wed, 28 Apr 2021 16:53:56 GMT
• Title: Schwinger's picture of quantum mechanics: 2-groupoids and symmetries
• Authors: Florio M. Ciaglia, Fabio Di Cosmo, Alberto Ibort, Giuseppe Marmo and Luca Schiavone
• Abstract summary: It is shown that, given a groupoid $Grightarrows Omega$ associated with a (quantum) system, there are two possible descriptions of its symmetries. On the other hand, taking advantage of the notion of group of bisections of a given groupoid, the global perspective allows to construct a group of symmetries out of a 2-groupoid. The latter notion allows to introduce an analog of the Wigner's theorem for quantum symmetries in the groupoid approach to Quantum Mechanics.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Starting from the groupoid approach to Schwinger's picture of Quantum Mechanics, a proposal for the description of symmetries in this framework is advanced.It is shown that, given a groupoid $G\rightrightarrows \Omega$ associated with a (quantum) system, there are two possible descriptions of its symmetries, one "microscopic", the other one "global".The microscopic point of view leads to the introduction of an additional layer over the grupoid $G$, giving rise to a suitable algebraic structure of 2-groupoid.On the other hand, taking advantage of the notion of group of bisections of a given groupoid, the global perspective allows to construct a group of symmetries out of a 2-groupoid.The latter notion allows to introduce an analog of the Wigner's theorem for quantum symmetries in the groupoid approach to Quantum Mechanics.

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